Interpreting Coefficients of -Regress- with multiple Interaction Terms

[Dan Rebenich]

Dear experienced STATA users,
Dear statisticians,

for a paper I am currently working on I utilise the following regression model:

regress y x1 x2 x3 x1#x2 x1#x3

where y = depvar, x1 - x3 = predictors and x1#x2 - x1#x3 are two interaction terms. Assuming I find a coefficient of 0.2 for x3, do I interpret this as:

A) The coefficient of the main effect of x3 being 0.2 for x1 = 0, holding x2 constant OR
B) The coefficient of the main effect of x3 being 0.2 for x1 = 0 and x2 = 0?

Reversely, assuming I find a coefficient of 0.4 for x1#x3, do I interpret this as:

C) The coefficient of the interaction effect of x1 and x3 being 0.4, holding x2 constant OR
D) The coefficient of the interaction effect of x1 and x3 being 0.4, for x2 = 0?

Thank you so much, help is greatly appreciated

[Clyde Schechter]

Assuming I find a coefficient of 0.2 for x3, do I interpret this as:

A) The coefficient of the main effect of x3 being 0.2 for x1 = 0, holding x2 constant OR
B) The coefficient of the main effect of x3 being 0.2 for x1 = 0 and x2 = 0?

Both are wrong, but B) is a bit closer to correct. What the coefficient of x3 represents is the marginal effect of x3 conditional on x2 = 0. While some people would refer to this as the "main" effect of x3, I think that terminology should be avoided because it has connotations of being some kind of privileged, overall effect of x3, whereas what it really is is just the effect conditional on one particular value of x2.

The value of x1 is not relevant here because x1 does not interact with x3.

do I interpret this as:

C) The coefficient of the interaction effect of x1 and x3 being 0.4, holding x2 constant OR
D) The coefficient of the interaction effect of x1 and x3 being 0.4, for x2 = 0?

C is correct. The value of x2 does have an effect on the marginal effect of x1, but it does not have any effect on the interaction between x1 and x3 in this model. If you had a three way interaction term x1#x2#x3, that would be a different story. Again, I don't like the terminology "interaction effect," though it is widely used, because it is not, strictly speaking an effect at all. It is a modification of an effect of a variable, a difference between effects of x1 depending on different values of x3.

If you are inclined towards calculus, the coefficient of x1 is a first-order partial derivative, whereas the coefficient of x1#whatever is a mixed second order partial derivative, which is a different species. Effects, strictly speaking, are always first order partial derivatives. (For discrete variables, substitute "difference" for "derivative".)

[Dan Rebenich]

What the coefficient of x3 represents is the marginal effect of x3 conditional on x2 = 0

I do understand your concerns regarding terminology and I will make sure to look into the term "marginal effect" more. I take away however, that the numerical coefficient provided to me is valid at x2 = 0, not merely for holding x2 constant.

Still:

The value of x1 is not relevant here because x1 does not interact with x3

This I do not understand. x1 in this model interacts with both x2 and x3, in different terms though. For any interpretation of a coefficient of x3, x1 hast to be 0 in my mind. Am I overlooking something?

C is correct. The value of x2 does have an effect on the marginal effect of x1, but it does not have any effect on the interaction between x1 and x3 in this model

Do you, per chance, know of any scientific publication that specifically explains this? I cannot seem to find it but I'd certainly love to read it!

Thank you so much, Clyde Schechter!

[Clyde Schechter]

The value of x1 is not relevant here because x1 does not interact with x3


This I do not understand. x1 in this model interacts with both x2 and x3, in different terms though. For any interpretation of a coefficient of x3, x1 hast to be 0 in my mind. Am I overlooking something?

My bad. I meant to write that because x2 does not interact with x3, the value of x2 does not impact the interpretation of the effect of x3: the effects of x3 come entirely from the coefficients of x3 itself, that of x1#x3, and the value of x1.

So let's go back to the beginning on this one. In a model -regress y x1 x2 x3 x1#x2 x1#x3-, the coefficient of x3 represents the marginal effect of x3 conditional on x1 = 0. For this statement, the value of x2 is completely irrelevant: it need not even be held constant. So this is actually closer to option A. I'm very sorry for my error and the confusion it caused.

[Dan Rebenich]

Thank you, Clyde! This has resolved my issue.

[George Ford]

y = b1*x1 + b2*x2 + b3*x3 + b4*(x1*x2) + b5*(x1*x3)

dy/dx1 = b1 + b4*x2 + b5*x3

dy/dx2 = b2 + b4*x1

dy/dx3 = b3 + b5*x1

Often, these are computed at the mean of the interaction term that is in the derivative. 0 is just one possible option, and 0 may not be an observed value. Other than make terms disappear, 0 is often not a useful value; it is simply one of many. The mean makes more sense. This is what Clyde is saying about the "main effect".

You can use margins/marginsplot to get a sense of things.

[Elliot Smith]

Dear all,

I wanted to reopen this thread as I find myself having troubles of interpretation for the same model that Dan had shown, but somehow I do not manage to clarify in my mind the interpretation of the coefficient of these models. So I saw in somebodys work the exact same model that Dan was showing, meaning it had the structure:

y = b1*x1 + b2*x2 + b3*x3 + b4*(x1*x2) + b5*(x1*x3)

In particular, it explored gender differences (x1=gender) in performance results in a exam (y = performance), conditional on number of days of preparation to the exam (x2= number of days) and subject type (x3 = stem/non-stem). Hence, the model was something like:

Model a: performance = b1*female + b2*STEM + b3* days + b4(female*STEM) + b5(female*days)

As I understand from manuals, and also following this post, then each coefficient can be interpreted as (I am bulleting my questios for easiness of answering):

• b1 = Female gap in performance in non-STEM subjects and 0 days preparation (or average days preparation when centered)?
• b2 = STEM gap for in performance for MALE students (ceteris parbus days of preparation) ?
• b3 = Effect of days preparation in performance for MALE students (ceteris parbus STEM vs non-STEM subjects) ?
• b4 = Difference in the gap in STEM for female students vs male (hence, FEMALE gap in STEM would be b4 + b2) ceteris parbus preparation days ?
• b5 = Difference in the effect of days for female vs male (hence, FEMALE slope would be b3 + b5) ceteris parbus subject type ?

Are these interpretations correct?

Additionally, an additional doubt emerges that is causing me a lot of confusion when looking at this model. So imagine I would only have this model instead:

Model b: performance = b1*female + b2*STEM + b3(female*STEM)

• As I understand,in model b, b3 could be interpreted not only as the "Difference in the gap in STEM for female students vs male" (as I interpreted for the model above), but it could also be interpreted as the difference in the gender gap in STEM vs non-STEM, so that the gender gap in STEM subjects would be b1 + b3. Is this correct?
• If this is so, going back to model A, how does this apply to the interpretation I made of b3? Would also reflect the gender gap of STEM vs non-STEM ceteris parbus days preparation? Except, if I am correct, it could not be estimated by adding it to b1, as b1 would describe the gap when there are 0 days of preparation, so they are not really comparable.

Thanks in advance for anybodys time spent responding to my questions. Best regards,

Elliot